The problem is to find an enumerable and canonical form of subgroups of $\mathbb{Z}_q^n$, where $\mathbb{Z}_q$ means the cyclic group $(\mathbb{Z}/ q\mathbb{Z}, +)$ for some positive integer $q$, and $\boxed{}^n \; (n \ge 2)$ means a composition of $n$ direct sums among groups.
If $q$ is prime, the problem seems to be easier. Anyway, I've made no progress so far.